2004
DOI: 10.1142/s0219891604000329
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Global Existence for Phase Transition Problems via a Variational Scheme

Abstract: We construct approximate solutions of the initial value problem for dynamical phase transition problems via a variational scheme in one space dimension. First, we deal with a local model of phase transition dynamics which contains second and third order spatial derivatives modeling the effects of viscosity and surface tension. Assuming that the initial data are periodic, we prove the convergence of approximate solutions to a weak solution which satisfies the natural dissipation inequality. We note that this re… Show more

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Cited by 2 publications
(3 citation statements)
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“…Since we are not interested in the case for positive but fixed value for ε we do not give the proof of this statement but focus on the limit process ε → 0. We note that the existence of weak solutions for (1.5), (1.6) under Assumption 4.1 has been shown in [39].…”
Section: Global Existence Of Classicalmentioning
confidence: 88%
See 1 more Smart Citation
“…Since we are not interested in the case for positive but fixed value for ε we do not give the proof of this statement but focus on the limit process ε → 0. We note that the existence of weak solutions for (1.5), (1.6) under Assumption 4.1 has been shown in [39].…”
Section: Global Existence Of Classicalmentioning
confidence: 88%
“…Systems with capillarity terms like D ε given by (1.7) have been analyzed by many authors ( [2,8,29,39]), also in the closely related case of liquid-vapour phase transitions in Van-der-Waals fluids ( [3,9,19,20,26,42]). In particular with the ε-scaling given by (1.5), (1.7) solutions of the Cauchy problem for (1.5) have been shown to converge to weak solutions of the Cauchy problem for the sharpinterface limit system (1.1) which contain dynamical phase boundaries ( [29]).…”
mentioning
confidence: 99%
“…For non-negative φ , the existence of weak solutions to system (1.1) for given initial values (on the spatial domain R) was proven for arbitrary Lipschitz continuous σ with a variational technique (see [13], Theorem 5.4) relying on the ideas in [4]. This result is extended in this contribution in various directions: initial boundary value problems are addressed, partly negative kernels are treated, and optimal regularity is proven under quite rough initial conditions.…”
Section: Introductionmentioning
confidence: 99%